Interferometric optical techniques are widely used to measure surface topography, optical thickness, flatness, and other geometric and refractive index properties of precision optical components such as glass substrates used in lithographic photomasks.
For example, to measure the surface profile of a measurement surface, one can use an interferometer to combine a measurement wavefront reflected from the measurement surface with a reference wavefront reflected from a reference surface to form an optical interference pattern. Spatial variations in the intensity profile of the optical interference pattern correspond to phase differences between the combined measurement and reference wavefronts caused by variations in the profile of the measurement surface relative to the reference surface. Phase-shifting interferometry (PSI) can be used to accurately determine the phase differences and the corresponding profile of the measurement surface.
With PSI, the optical interference pattern is recorded for each of multiple phase-shifts between the reference and measurement wavefronts to produce a series of optical interference patterns that span at least a full cycle of optical interference (e.g., from constructive, to destructive, and back to constructive interference). The optical interference patterns define a series of intensity values for each spatial location of the pattern, wherein each series of intensity values has a sinusoidal dependence on the phase-shifts with a phase-offset equal to the phase difference between the combined measurement and reference wavefronts for that spatial location. Using numerical techniques known in the art, the phase-offset for each spatial location is extracted from the sinusoidal dependence of the intensity values to provide a profile of the measurement surface relative the reference surface. Such numerical techniques are generally referred to as phase-shifting algorithms.
The phase-shifts in PSI can be produced by changing the optical path length from the measurement surface to the interferometer relative to the optical path length from the reference surface to the interferometer. For example, the reference surface can be moved relative to the measurement surface. In addition to such mechanical phase-shifting, phase shifts can be introduced by electro-optic or acousto-optical modulation. Furthermore, the phase-shifts can be introduced for a constant, non-zero optical path difference by changing the wavelength of the measurement and reference wavefronts. The latter application is known as wavelength tuning PSI and is described, e.g., in U.S. Pat. No. 4,594,003 to G. E. Sommargren.
One example of a phase-shifting interferometer is a Fizeau interferometer illuminated with a coherent source, such as a laser. For example, a test surface of arbitrary shape is imaged with a Fizeau interferometer capable of producing a controlled phase shift along a Z axis for a PSI acquisition with the chosen algorithm. The optical system is aligned along the Z axis and the surface is imaged onto a camera so each pixel corresponds to a unique position in the XY plane. A laser beam is directed towards a reference surface and the test surface, and the interference between the light beams reflected from the test and reference surfaces are sampled as a function of phase shift and subsequently analyzed with a PSI algorithm to extract the test surface phase map, which is converted into physical units using the known wavelength of the laser beam. A conventional PSI algorithm assumes a constant scanning motion (i.e., constant velocity).
If the scanning motion is not uniform, errors in the measured surface profile occur. Unfortunately, it is often the case that the scanning motion in PSI is not uniform. This can occur due to nonlinear motions of the scanning mechanism, or through vibrations that act on each component of the interferometer differently.